16.5

1. This section wasn't extremely difficult, but I would like to investigate what properties inverses in some field have when they are considered in an enclosing structure, like the ring Z.

2. We've talked a couple times about how there is no real notion of smallness when it comes to points on the curve E, and how it is for that reason that we can't use the same methods to attack the discrete logarithm problem for elliptic curves. At first I was tempted to say that we picked the wrong analogy and that comparing it to a discrete logarithm was guiding our intuition in the wrong direction, but I suppose it is the same problem if we used multiplicative notation instead of additive. Anyway, my point is that perhaps there is someway to define smallness based on certain properties, etc.

16.4

1. The most difficult part for me was realizing that one of the more significant issues in GF(2) and GF(4) (maybe GF(2^n) in general?) is finding the additive inverse of an element and not adding two elements together, but I guess it turns out to be the same thing. It takes an extra step.

2. Even though fields in Z_2 are a more natural fit for computers, does that mean that fields Z_2[x] are also? (This next part is all "if I'm not mistaken".) In the text, only a few lines were dealt with: vertical, horizontal, and the line y=x. A line could have the slope of any element in the group. In GF(4), names were given to each element, so I suppose in, say, GF(256), even though most of the elements may have the ring element x in their representation, this is a different x than the one appearing in y=mx+b, so if considering the line through (0,1) and (1, x^2+x+1) on some hypothetical curve, we would obtain y=(x^2+x+1)x. Is that y=x^3+x^2+x? In other words, are the x's the same? I'm leaning towards yes.

16.3

1. I had a question about the p-1 analogue for factoring but in writing it, I saw the answer. However, why the process isolates the behavior of one of the factors is still a mystery to me. That and the entire second half of the reading.

2. It's interesting to me that choosing a random curve (mod n) results in essentially a random number of points, around n, on the curve. Since there is a practical limit to the size of numbers that can be factored using this method, I'm going to assume that the time complexity goes up exponentially with the number of digits (so essentially linearly with the value of the number). I have nothing of worth to say about elliptic curves themselves, unfortunately.

16.2

1. Pardon my ignorance, but how can we take a derivative of a function (implicit or not) when there is no (obvious) notion of continuity. Don't we use difference quotients or something? Is that the same thing in this case?

2. I am consistently amazed at the connections and applications that seemingly obscure branches of math have. First, the recognition that many of the results of, say, topology come not from a construct itself (say, the complex numbers), but from the properties that construct has (field properties). It then allows one to interpret things like fractions in finite fields where all members are representable by integers. More pertinently, the investigation of arc length of ellipses led to a more intense study of elliptic curves which turn out to be extremely fascinating. I must say apologetically that I have nothing of interest to say about elliptic curves over Z mod n. This is all new to me.

16.1

1. Why do we not have to consider a quadratic term in the x's? Is it possible to write any elliptic curve without that term by using a method similar to completing the square? Also, isn't it possible to choose two points so that a third point isn't intersected? What then?

2. I have never been exposed to elliptic curves before, at least not in an expository setting. Thinking about this for a moment, I bet that any elliptic curve with rational coefficients can be obtained through a rational map with integer coefficients. I have discovered a truly remarkable proof for this, but it is too large to put in this blog post. Seriously though, it is pretty incredible that such seemingly unrelated fields (algebra and analysis? it's all numbers anyway) could be linked in such an elegant way. And to think that that link somehow proves that x^n+y^n=z^n has no integer solutions for n>2. It's amazing.

2.12

1. I'm not going to pretend I know exactly how the Enigma works. I grok the permutations and stuff, but I don't know how the mechanics of the machine works. Which leads us to...

2. At this time, computation was still in its relative infancy. Whether Alan Turing's involvement at Bletchley Park abetted his role as the father of modern computation or vice versa, I'm uncertain, but the timing is interesting. The reason I bring this up is because one of the weaknesses was in their "error correcting codes" which I think fall under the umbrella of coding theory. Since radio transmission was prone to error, they introduced direct redundancy which weakened their security. It is exactly this situation that led Claude Shannon to develop the mathematical theory of communication, information theory, which explores the physical limits of information transfer through a noisy channel. Physical might not be the right word. Anyway, what I am spending far too long to get at is it's pretty amazing that they could reverse engineer such a convoluted device. We have a hard enough time seeing a string of numbers and determining how it was encrypted, but we have at least some idea. It would be very easy for the Germans to obscure their messages in simple ways so that it would be harder for a cracker to realize they were on the right track. It really does make you think about 2 Nephi 31, to a point.

19.3

1. Aside from quantum anything, the most difficult part for me is the Fourier transform. I know that it transforms things from the time or spatial domain to the frequency domain, but despite my continued readings on the subject, I just can't grasp it.

2. Quantum computing changes the game, but I think it is not the panacea that many make it out to be (and the author agrees with me on this point). If I'm not mistaken, there are provably secure encryption models based on the behavior of quantum systems. Of course it is in the best interest of governments and institutions to keep their information secure, but do we expect that overcoming the physical limitations of quantum computing will allow us to use an arbitrarily large number of qubits (or maybe a practically large)? If not, we can always stay ahead of the tech with longer and longer keys, right?